1. Field of the Invention
The present invention relates generally to the fields of signal processing. More particularly, it concerns methods and systems for signal processing with fast S-transforms.
2. Description of the Related Art
The ability to detect the frequency content of a signal is a desirable capability in diverse applications from electrical engineering to neuroscience. Many transforms for accomplishing this task by transforming signals of one or more dimensions into frequency or frequency-analogue spaces are known to those of skill in the art.
While the original Fourier transform (FT) is an extremely important signal and image analysis tool, it assumes that a signal is stationary, i.e., that the frequency content is constant at all times in a signal, or at all locations in an image. Since most interesting signals are non-stationary, a series of techniques have been developed to characterize signals with dynamic frequency content. These are the foundation of the field of time-frequency analysis.
A simple approach to the problem of non-stationary signal analysis is the short-time Fourier transform (STFT). In this technique changes in frequency over time are captured using a window function to provide time localization. However, the choice of window function represents a compromise. Narrower windows provide better time resolution but poorer frequency resolution, while wider windows provide the converse. Ideally, the window width is chosen to produce the best representation of particular features of interest in the signal, but this requires a priori knowledge.
The wavelet transform (WT), which has been applied to a wide variety of signal processing problems, improves on the STFT by introducing the concept of progressive resolution. The WT provides the equivalent of finer time resolution at high frequencies and finer frequency resolution at low frequencies. However, the WT does not measure frequency but only an analogue, called scale. Additionally, the WT provides either no phase information, or phase measurements which are all relative to different local references. This is in contrast to the conventional concept of phase, as provided by the FT, where all phase measurements are relative to a global reference.
The S-transform (ST) exhibits the globally referenced phase and true frequency measurements of the FT and STFT, as well as the progressive resolution of the WT. This combination of desirable features have shown promise in a wide variety of applications, including, for example, detecting abnormalities in the heart, identifying genetic abnormalities in brain tumors, analyzing electroencephalograms, transmitting medical images, characterizing the behavior of liquid crystals, detecting disturbances in electrical power distribution networks monitoring high altitude wind patterns, and detecting gravitational waves. However, while very efficient methods have been developed for calculating the FT and WT, the computational demands of the ST have limited its utility.